**10. Axes problem**

268 Recent Advances in Crystallography

(**W**p,**w**). Since (**W**p,**w**)*<sup>k</sup>*

given in Table 4.

**9. Origin problem** 

relative to two origins.

The orthogonal splitting of a translation part **w** depends on the axis direction [*uvw*]. The same results are obtained for (**W**,**w**) and (-**W**,**w**), thus calculations can be reduced to the pair

component of the translation, and in consequence also the location component **w**l = **w**-**w**g. In the presented procedure such decomposition is also valuable for a rotoinversion operation. As usual, for a reflection the numerical values of **w**g and **w**l must be interchanged. The central problem of deriving the point **x**c (an invariant relative to a reduced operation and closest to the origin) is to find such point on the [*uvw*] lattice vector or on the (*hkl*) plane for a rotation and rotoinversion in the first case and for a reflection in the latter. Geometrical considerations showed that **x**c are rather simply calculated from **w**l according to relations

The derivation of **x**c ends the calculation needed to describe the space symmetry operation. But vectors **x**c in the case of reflections and **w**g in the case of rotations or rotoinversions are parallel to [*uvw*] and may be represented by a single ratio r. Thus, an intrinsic component **w**<sup>g</sup> and a shift vector (orthogonal to the geometric element) are presented by four simple ratios. An asterisk mark serves as a separation mark between vectors and thus a dual symbol of

In the non-symmorphic space groups the symmetry elements are not constrained to pass through the origin. The selection of a reasonable origin for a coordinate system relative to non-intersecting symmetry elements is not unique. The symbol of space-group type, like Hermann-Mauguin symbol, fixes in space only relative positions of symmetry elements. Absolute positions need complete translational parts in the space-group generators. The presented technique of space group derivation based on predefined generators favours one and the only one origin for each space group, even if this group is tabulated in ITA83

Finding a transformation between two descriptions of the same space group differing only by the origin shift is arithmetically at least cumbersome. In this case, similarly like in other space-group considerations, the geometric information is very practical. A typical way of resolving the mentioned problem consists of a geometrical interpretation of the symmetry matrices in both descriptions and of deduction of the transformation from the diagram of symmetry elements in ITA83. Such analysis is impossible for a non-conventional space

Since the dual symbols described in the preceding section are easy to obtain, they should be routinely derived together with the space group generation. Their role in the origin control is rather evident, but we illustrate this feature by means of an example. Let column 2 of Table 9 lists the dual symbols of the *P*42/*nnm* operations obtained from the generators presented in Table 8. The items 1, 7, 9, 15 have reduced **x**c parts. It is visible that the origin is located at the inversion point, the intersection of 2[110] and m[110]. A full symmetry of the

group description, but in every case may be carried out on dual symbols.

origin is 2/m (this is the second origin choice tabulated in ITA83).

space-group operation takes the form of ±n ± [*uvw*](*hkl*) **w**g\***x**c.

= (**I**,*k***wg**), the *k*-th part of the resulting translation defines the intrinsic

A rigorous classification of space groups, that is their specific descriptions, into space-group type can be given in an algebraic or a geometric way. Typically, the matrix algebra and the group-theoretical approach is preferred. For this classification, each space group description is referred to a primitive base and an origin. Two space groups **G** and **G'** belong to the same space-group type if a transformation pair **P**, **p** exists, for which the 3x3 matrix has integral elements with det(**P**) = 1 and the **p** vector consists of three real numbers, such that **G** is transformed into **G**' (see, Wondratschek in ITA83). This definition is very simple, but finding the transformation between two sets of matrices may be a real challenge.

Let's modify the above equivalence definition for the practical purposes. Now **G'** means a conventionally described *space-group type*, represented by a unique set of generators or symmetry matrices. **G** is still referred to a primitive base. **G** belongs to the space-group type **G'** if a transformation pair **P**, **p** exists, for which the 3x3 matrix has integral elements with det(**P**) = 1,2,3 or 4 and the **p** vector consists of three real numbers, such that **G** is transformed into **G**'. The first step in determining the type of group **G** is to refer it to a centred Bravais base by a proper selection of coordinate axes. It is simple with the help of dual symbols what will be illustrated by the group description TSG = *I*4122 (1,0,0; 0,1,0; ½,1/2,1/2) (1/4,1/4,0) given in Table 10.


**Table 10.** Symmetry matrices and dual symbols of the group description TSG = *I*4122 (1,0,0; 0,1,0; ½,1/2,1/2) (1/4,1/4,0)

Items 1,2,3 and 4 define symmetry axis 41 parallel to [1�1�2]. Matrices (5,6) and (7,8) describe two pairs of orthogonally oriented twofold axes, according to property *u*1*h*2 + *v*1*k*2 + *w*1*l*2 = *u*2*h*<sup>1</sup> + *v*2*k*1 + *w*2*l*1 = 0 between the corresponding splitting indices. A similar test shows the orthogonality between 41 and all twofold axes. Thus, two orthogonal bases and two transformation matrices may be constructed from [*uvw*] indices. The first transformation matrix **P** with columns [100], [010], [1�1�2] has det(**P**) = 2 and leads to *I*-centred basis. The equivalent F-centred Bravais tetragonal cell is involved with P in the form [11�0], [110] and [1�1�2]. The application of the first transformation matrix is equivalent to the selection of the conventional axes for the tetragonal space group.

Moreover, the arithmetic type 422*I* of the analysed group is determined. From the predefined data one can find that only two space-group types, namely the groups with sequence numbers 97 and 98, belong to this arithmetic type. Group 97 is symmorphic. Thus, 41 axis points directly onto type 98 space group. The origin shift **p** may be determined by comparing points **x**c in dual symbols of the space-group type generators and symmetry operations of the analysed group with identical matrix parts. The generators of group type *I*4122 are characterized by dual symbols as: 4+[001] 1/4 \* -1/4,1/4,0 and 2 [010] \* 1/4,0,3/8. Items 3 and 5 from Table 10 transformed according to **P** matrix give 4+[001] 1/4 \* 0,1/2,0 and 2 [010] 1/2 \* 1/4,0,1/8. The origin shift **p** = -1/4,-1/4,0 predicted from the 4� � operation equals both descriptions.

The analysed example does not attempt to be a kind of algorithmic approach, but was included to show the advantages of representing symmetry operations in both forms, as a symmetry matrix and as a dual symbol.
